def test_roots2():
"""Just test that calculating these roots does not hang
(final result is not checked)
"""
a, b, c, d, x = symbols("a b c d x")
f1 = x**2*c + (a/b) + x*c*d - a
f2 = x**2*(a + b*(c-d)*a) + x*a*b*c/(b*d-d) + (a*d-c/d)
assert roots(f1, x).values() == [1, 1]
assert roots(f2, x).values() == [1, 1]
def test_roots2():
"""Just test that calculating these roots does not hang
(final result is not checked)
"""
a, b, c, d, x = symbols("a b c d x")
f1 = x**2*c + (a/b) + x*c*d - a
f2 = x**2*(a + b*(c-d)*a) + x*a*b*c/(b*d-d) + (a*d-c/d)
assert roots(f1, x).values() == [1, 1]
assert roots(f2, x).values() == [1, 1]
def test_roots():
assert roots(1, x) == {}
assert roots(x, x) == {S.Zero: 1}
assert roots(x**9, x) == {S.Zero: 9}
assert roots(2*x+1, x) == {-S.Half: 1}
assert roots((2*x+1)**2, x) == {-S.Half: 2}
assert roots((2*x+1)**5, x) == {-S.Half: 5}
assert roots((2*x+1)**10, x) == {-S.Half: 10}
assert roots(x**4 - 1, x) == {I: 1, S.One: 1, -S.One: 1, -I: 1}
assert roots((x**4 - 1)**2, x) == {I: 2, S.One: 2, -S.One: 2, -I: 2}
assert roots(((2*x-3)**2).expand(), x) == { Rational(3,2): 2}
assert roots(((2*x+3)**2).expand(), x) == {-Rational(3,2): 2}
assert roots(((2*x-3)**3).expand(), x) == { Rational(3,2): 3}
assert roots(((2*x+3)**3).expand(), x) == {-Rational(3,2): 3}
assert roots(((2*x-3)**5).expand(), x) == { Rational(3,2): 5}
assert roots(((2*x+3)**5).expand(), x) == {-Rational(3,2): 5}
assert roots(((a*x-b)**5).expand(), x) == { b/a: 5}
assert roots(((a*x+b)**5).expand(), x) == {-b/a: 5}
assert roots(x**4-2*x**2+1, x) == {S.One: 2, -S.One: 2}
assert roots(x**6-4*x**4+4*x**3-x**2, x) == \
{S.One: 2, -1 - sqrt(2): 1, S.Zero: 2, -1 + sqrt(2): 1}
assert roots(x**8-1, x) == {
2**S.Half/2 + I*2**S.Half/2: 1,
2**S.Half/2 - I*2**S.Half/2: 1,
-2**S.Half/2 + I*2**S.Half/2: 1,
-2**S.Half/2 - I*2**S.Half/2: 1,
S.One: 1, -S.One: 1, I: 1, -I: 1
}
assert roots((a+b+c)*x + a+b+c+d, x) == \
{ (-a-b-c-d) / (a+b+c) : 1 }
assert roots(x**3+x**2-x+1, x, cubics=False) == {}
r1_2, r1_3, r1_9, r4_9, r19_27 = [ Rational(*r) \
for r in ((1,2), (1,3), (1,9), (4,9), (19,27)) ]
assert roots(x**3+x**2-x+1, x, cubics=True) in [
{
-r1_3 - (r19_27 + r1_9*3**r1_2*11**r1_2)**r1_3 - \
r4_9*(r19_27 + r1_9*3**r1_2*11**r1_2)**(-r1_3): 1,
-r1_3 + r1_2*(r19_27 + r1_9*3**r1_2*11**r1_2)**r1_3 + \
r4_9/(r1_2 + r1_2*I*3**r1_2)*(r19_27 + r1_9*3**r1_2*11**r1_2)**(-r1_3) + \
r1_2*I*3**r1_2*(r19_27 + r1_9*3**r1_2*11**r1_2)**r1_3: 1,
-r1_3 + r1_2*(r19_27 + r1_9*3**r1_2*11**r1_2)**r1_3 + \
r4_9/(r1_2 - r1_2*I*3**r1_2)*(r19_27 + r1_9*3**r1_2*11**r1_2)**(-r1_3) - \
r1_2*I*3**r1_2*(r19_27 + r1_9*3**r1_2*11**r1_2)**r1_3: 1,
},
{
-r1_3 - (r19_27 + r1_9*3**r1_2*11**r1_2)**r1_3 - \
r4_9*(r19_27 + r1_9*3**r1_2*11**r1_2)**(-r1_3): 1,
-r1_3 + r1_2*(r19_27 + r1_9*3**r1_2*11**r1_2)**r1_3 - \
r4_9/(-r1_2 - r1_2*I*3**r1_2)*(r19_27 + r1_9*3**r1_2*11**r1_2)**(-r1_3) + \
r1_2*I*3**r1_2*(r19_27 + r1_9*3**r1_2*11**r1_2)**r1_3: 1,
-r1_3 + r1_2*(r19_27 + r1_9*3**r1_2*11**r1_2)**r1_3 + \
r4_9/(r1_2 - r1_2*I*3**r1_2)*(r19_27 + r1_9*3**r1_2*11**r1_2)**(-r1_3) - \
r1_2*I*3**r1_2*(r19_27 + r1_9*3**r1_2*11**r1_2)**r1_3: 1,
},
]
f = (x**2+2*x+3).subs(x, 2*x**2 + 3*x).subs(x, 5*x-4)
r1_2, r2_25, r13_20, r1_100 = [ Rational(*r) \
for r in ((1,2), (2,25), (13,20), (1,100)) ]
assert roots(f, x) == {
r13_20 + r1_2*(r1_100 - r2_25*I*2**r1_2)**r1_2: 1,
r13_20 - r1_2*(r1_100 - r2_25*I*2**r1_2)**r1_2: 1,
r13_20 + r1_2*(r1_100 + r2_25*I*2**r1_2)**r1_2: 1,
r13_20 - r1_2*(r1_100 + r2_25*I*2**r1_2)**r1_2: 1,
}
p = Poly(z**3 + (-2 - y)*z**2 + (1 + 2*y - 2*x**2)*z - y + 2*x**2, z)
assert roots(p) == {
S.One: 1,
S.Half + S.Half*y + S.Half*(1 - 2*y + y**2 + 8*x**2)**S.Half: 1,
S.Half + S.Half*y - S.Half*(1 - 2*y + y**2 + 8*x**2)**S.Half: 1,
}
assert roots(x**3 + 2*x**2 + 4*x + 8, x) == {}
assert roots(x**3 + 2*x**2 + 4*x + 8, x, cubics=True) == \
{-2*I: 1, 2*I: 1, -Integer(2): 1}
assert roots(a*b*c*x**3 + 2*x**2 + 4*x + 8, x) == {}
assert roots(a*b*c*x**3 + 2*x**2 + 4*x + 8, x, cubics=True) != {}
assert roots(x**4-1, x, domain='Z') == {S.One: 1, -S.One: 1}
assert roots(x**4-1, x, domain='I') == {I: 1, -I: 1}
assert roots((x-1)*(x+1), x) == {S.One: 1, -S.One: 1}
assert roots((x-1)*(x+1), x, predicate=lambda r: r.is_positive) == {S.One: 1}
assert roots(x**4-1, x, domain='Z', multiple=True) == [S.One, -S.One]
assert roots(x**4-1, x, domain='I', multiple=True) in ([I, -I], [-I, I])
assert roots(x**3, x, multiple=True) == [S.Zero, S.Zero, S.Zero]
assert roots(1234, x, multiple=True) == []
def test_roots():
assert roots(1, x) == {}
assert roots([1]) == {}
assert roots(x) == {S.Zero: 1}
assert roots(x, x) == {S.Zero: 1}
uv = Poly(x, x)
mv = Poly(x + y, x, y)
assert roots(uv) == {S.Zero: 1}
assert roots(uv, x) == {S.Zero: 1}
assert roots(mv, x) == {-y: 1}
raises(SymbolsError,"roots([1],x)")
raises(SymbolsError,"roots(x+y, x, y)")
raises(SymbolsError,"roots(uv, x, y)")
raises(SymbolsError,"roots(mv, x, y)")
raises(MultivariatePolyError,"roots(mv)")
assert roots(x**9, x) == {S.Zero: 9}
assert roots(((x-2)*(x+3)*(x-4)).expand(), x) == {-S(3): 1, S(2): 1, S(4): 1}
assert roots(2*x+1, x) == roots([2, 1])
assert roots([2, 1]) == {-S.Half: 1}
assert roots((2*x+1)**2, x) == {-S.Half: 2}
assert roots((2*x+1)**5, x) == {-S.Half: 5}
assert roots((2*x+1)**10, x) == {-S.Half: 10}
assert roots(x**4 - 1, x) == {I: 1, S.One: 1, -S.One: 1, -I: 1}
assert roots((x**4 - 1)**2, x) == {I: 2, S.One: 2, -S.One: 2, -I: 2}
assert roots(((2*x-3)**2).expand(), x) == { Rational(3,2): 2}
assert roots(((2*x+3)**2).expand(), x) == {-Rational(3,2): 2}
assert roots(((2*x-3)**3).expand(), x) == { Rational(3,2): 3}
assert roots(((2*x+3)**3).expand(), x) == {-Rational(3,2): 3}
assert roots(((2*x-3)**5).expand(), x) == { Rational(3,2): 5}
assert roots(((2*x+3)**5).expand(), x) == {-Rational(3,2): 5}
assert roots(((a*x-b)**5).expand(), x) == { b/a: 5}
assert roots(((a*x+b)**5).expand(), x) == {-b/a: 5}
assert roots(x**4-2*x**2+1, x) == {S.One: 2, -S.One: 2}
assert roots(x**6-4*x**4+4*x**3-x**2, x) == \
{S.One: 2, -1 - sqrt(2): 1, S.Zero: 2, -1 + sqrt(2): 1}
assert roots(x**8-1, x) == {
2**S.Half/2 + I*2**S.Half/2: 1,
2**S.Half/2 - I*2**S.Half/2: 1,
-2**S.Half/2 + I*2**S.Half/2: 1,
-2**S.Half/2 - I*2**S.Half/2: 1,
S.One: 1, -S.One: 1, I: 1, -I: 1
}
assert roots(-2016*x**2 - 5616*x**3 - 2056*x**4 + 3324*x**5 + 2176*x**6 \
- 224*x**7 - 384*x**8 - 64*x**9, x) == {S(0): 2, -S(2): 2, S(2): 1, -S(7)/2: 1,\
-S(3)/2: 1, -S(1)/2: 1, S(3)/2: 1}
assert roots((a+b+c)*x + a+b+c+d, x) == \
{ (-a-b-c-d) / (a+b+c) : 1 }
assert roots(x**3+x**2-x+1, x, cubics=False) == {}
assert roots(((x-2)*(x+3)*(x-4)).expand(), x, cubics=False) == {-S(3): 1, S(2): 1, S(4): 1}
assert roots(((x-2)*(x+3)*(x-4)*(x-5)).expand(), x, cubics=False) == \
{-S(3): 1, S(2): 1, S(4): 1, S(5): 1}
assert roots(x**3 + 2*x**2 + 4*x + 8, x) == {-S(2): 1, -2*I: 1, 2*I: 1}
assert roots(x**3 + 2*x**2 + 4*x + 8, x, cubics=True) == \
{-2*I: 1, 2*I: 1, -S(2): 1}
assert roots((x**2 - x)*(x**3 + 2*x**2 + 4*x + 8), x ) == \
{S(1): 1, S(0): 1, -S(2): 1, -2*I: 1, 2*I: 1}
r1_2, r1_3, r1_9, r4_9, r19_27 = [ Rational(*r) \
for r in ((1,2), (1,3), (1,9), (4,9), (19,27)) ]
assert set(tmp.evalf() for tmp in roots(x**3+x**2-x+1, x, cubics=True)) == \
set(tmp.evalf() for tmp in [
-r1_3 - (r19_27 + r1_9*3**r1_2*11**r1_2)**r1_3 - \
r4_9*(r19_27 + r1_9*3**r1_2*11**r1_2)**(-r1_3),
-r1_3 + r1_2*(r19_27 + r1_9*3**r1_2*11**r1_2)**r1_3 + \
r4_9/(r1_2 + r1_2*I*3**r1_2)*(r19_27 + r1_9*3**r1_2*11**r1_2)**(-r1_3) + \
r1_2*I*3**r1_2*(r19_27 + r1_9*3**r1_2*11**r1_2)**r1_3,
-r1_3 + r1_2*(r19_27 + r1_9*3**r1_2*11**r1_2)**r1_3 + \
r4_9/(r1_2 - r1_2*I*3**r1_2)*(r19_27 + r1_9*3**r1_2*11**r1_2)**(-r1_3) - \
r1_2*I*3**r1_2*(r19_27 + r1_9*3**r1_2*11**r1_2)**r1_3
])
f = (x**2+2*x+3).subs(x, 2*x**2 + 3*x).subs(x, 5*x-4)
r1_2, r13_20, r1_100 = [ Rational(*r) \
for r in ((1,2), (13,20), (1,100)) ]
ans = {
r13_20 + r1_100*(25 - 200*I*2**r1_2)**r1_2: 1,
r13_20 - r1_100*(25 - 200*I*2**r1_2)**r1_2: 1,
r13_20 + r1_100*(25 + 200*I*2**r1_2)**r1_2: 1,
r13_20 - r1_100*(25 + 200*I*2**r1_2)**r1_2: 1,
}
assert set(tmp.evalf() for tmp in roots(f, x).keys()) == \
set(tmp.evalf() for tmp in ans.keys())
p = Poly(z**3 + (-2 - y)*z**2 + (1 + 2*y - 2*x**2)*z - y + 2*x**2, z)
ans = {
S.One: 1,
S.Half + S.Half*y + S.Half*(1 - 2*y + y**2 + 8*x**2)**S.Half: 1,
S.Half + S.Half*y - S.Half*(1 - 2*y + y**2 + 8*x**2)**S.Half: 1,
}.keys()
assert all(tmp.expand() in ans for tmp in roots(p).keys())
assert roots(a*b*c*x**3 + 2*x**2 + 4*x + 8, x, cubics=False) == {}
assert roots(a*b*c*x**3 + 2*x**2 + 4*x + 8, x, cubics=True) != {}
assert roots(x**4-1, x, domain='Z') == {S.One: 1, -S.One: 1}
assert roots(x**4-1, x, domain='I') == {I: 1, -I: 1}
assert roots((x-1)*(x+1), x) == {S.One: 1, -S.One: 1}
assert roots((x-1)*(x+1), x, predicate=lambda r: r.is_positive) == {S.One: 1}
assert roots(x**4-1, x, domain='Z', multiple=True) == [S.One, -S.One]
assert roots(x**4-1, x, domain='I', multiple=True) in ([I, -I], [-I, I])
assert roots(x**3, x, multiple=True) == [S.Zero, S.Zero, S.Zero]
assert roots(1234, x, multiple=True) == []
def test_roots():
assert roots(1, x) == {}
assert roots(x, x) == {S.Zero: 1}
assert roots(x**9, x) == {S.Zero: 9}
assert roots(((x-2)*(x+3)*(x-4)).expand(), x) == {-S(3): 1, S(2): 1, S(4): 1}
assert roots(2*x+1, x) == {-S.Half: 1}
assert roots((2*x+1)**2, x) == {-S.Half: 2}
assert roots((2*x+1)**5, x) == {-S.Half: 5}
assert roots((2*x+1)**10, x) == {-S.Half: 10}
assert roots(x**4 - 1, x) == {I: 1, S.One: 1, -S.One: 1, -I: 1}
assert roots((x**4 - 1)**2, x) == {I: 2, S.One: 2, -S.One: 2, -I: 2}
assert roots(((2*x-3)**2).expand(), x) == { Rational(3,2): 2}
assert roots(((2*x+3)**2).expand(), x) == {-Rational(3,2): 2}
assert roots(((2*x-3)**3).expand(), x) == { Rational(3,2): 3}
assert roots(((2*x+3)**3).expand(), x) == {-Rational(3,2): 3}
assert roots(((2*x-3)**5).expand(), x) == { Rational(3,2): 5}
assert roots(((2*x+3)**5).expand(), x) == {-Rational(3,2): 5}
assert roots(((a*x-b)**5).expand(), x) == { b/a: 5}
assert roots(((a*x+b)**5).expand(), x) == {-b/a: 5}
assert roots(x**4-2*x**2+1, x) == {S.One: 2, -S.One: 2}
assert roots(x**6-4*x**4+4*x**3-x**2, x) == \
{S.One: 2, -1 - sqrt(2): 1, S.Zero: 2, -1 + sqrt(2): 1}
assert roots(x**8-1, x) == {
2**S.Half/2 + I*2**S.Half/2: 1,
2**S.Half/2 - I*2**S.Half/2: 1,
-2**S.Half/2 + I*2**S.Half/2: 1,
-2**S.Half/2 - I*2**S.Half/2: 1,
S.One: 1, -S.One: 1, I: 1, -I: 1
}
assert roots(-2016*x**2 - 5616*x**3 - 2056*x**4 + 3324*x**5 + 2176*x**6 \
- 224*x**7 - 384*x**8 - 64*x**9, x) == {S(0): 2, -S(2): 2, S(2): 1, -S(7)/2: 1,\
-S(3)/2: 1, -S(1)/2: 1, S(3)/2: 1}
assert roots((a+b+c)*x + a+b+c+d, x) == \
{ (-a-b-c-d) / (a+b+c) : 1 }
assert roots(x**3+x**2-x+1, x, cubics=False) == {}
assert roots(((x-2)*(x+3)*(x-4)).expand(), x, cubics=False) == {-S(3): 1, S(2): 1, S(4): 1}
assert roots(((x-2)*(x+3)*(x-4)*(x-5)).expand(), x, cubics=False) == \
{-S(3): 1, S(2): 1, S(4): 1, S(5): 1}
assert roots(x**3 + 2*x**2 + 4*x + 8, x) == {-S(2): 1, -2*I: 1, 2*I: 1}
assert roots(x**3 + 2*x**2 + 4*x + 8, x, cubics=True) == \
{-2*I: 1, 2*I: 1, -S(2): 1}
assert roots((x**2 - x)*(x**3 + 2*x**2 + 4*x + 8), x ) == \
{S(1): 1, S(0): 1, -S(2): 1, -2*I: 1, 2*I: 1}
r1_2, r1_3, r1_9, r4_9, r19_27 = [ Rational(*r) \
for r in ((1,2), (1,3), (1,9), (4,9), (19,27)) ]
assert roots(x**3+x**2-x+1, x, cubics=True) in [
{
-r1_3 - (r19_27 + r1_9*3**r1_2*11**r1_2)**r1_3 - \
r4_9*(r19_27 + r1_9*3**r1_2*11**r1_2)**(-r1_3): 1,
-r1_3 + r1_2*(r19_27 + r1_9*3**r1_2*11**r1_2)**r1_3 + \
r4_9/(r1_2 + r1_2*I*3**r1_2)*(r19_27 + r1_9*3**r1_2*11**r1_2)**(-r1_3) + \
r1_2*I*3**r1_2*(r19_27 + r1_9*3**r1_2*11**r1_2)**r1_3: 1,
-r1_3 + r1_2*(r19_27 + r1_9*3**r1_2*11**r1_2)**r1_3 + \
r4_9/(r1_2 - r1_2*I*3**r1_2)*(r19_27 + r1_9*3**r1_2*11**r1_2)**(-r1_3) - \
r1_2*I*3**r1_2*(r19_27 + r1_9*3**r1_2*11**r1_2)**r1_3: 1,
},
{
-r1_3 - (r19_27 + r1_9*3**r1_2*11**r1_2)**r1_3 - \
r4_9*(r19_27 + r1_9*3**r1_2*11**r1_2)**(-r1_3): 1,
-r1_3 + r1_2*(r19_27 + r1_9*3**r1_2*11**r1_2)**r1_3 - \
r4_9/(-r1_2 - r1_2*I*3**r1_2)*(r19_27 + r1_9*3**r1_2*11**r1_2)**(-r1_3) + \
r1_2*I*3**r1_2*(r19_27 + r1_9*3**r1_2*11**r1_2)**r1_3: 1,
-r1_3 + r1_2*(r19_27 + r1_9*3**r1_2*11**r1_2)**r1_3 + \
r4_9/(r1_2 - r1_2*I*3**r1_2)*(r19_27 + r1_9*3**r1_2*11**r1_2)**(-r1_3) - \
r1_2*I*3**r1_2*(r19_27 + r1_9*3**r1_2*11**r1_2)**r1_3: 1,
},
]
f = (x**2+2*x+3).subs(x, 2*x**2 + 3*x).subs(x, 5*x-4)
r1_2, r13_20, r1_100 = [ Rational(*r) \
for r in ((1,2), (13,20), (1,100)) ]
assert roots(f, x) == {
r13_20 + r1_100*(25 - 200*I*2**r1_2)**r1_2: 1,
r13_20 - r1_100*(25 - 200*I*2**r1_2)**r1_2: 1,
r13_20 + r1_100*(25 + 200*I*2**r1_2)**r1_2: 1,
r13_20 - r1_100*(25 + 200*I*2**r1_2)**r1_2: 1,
}
p = Poly(z**3 + (-2 - y)*z**2 + (1 + 2*y - 2*x**2)*z - y + 2*x**2, z)
assert roots(p) == {
S.One: 1,
S.Half + S.Half*y + S.Half*(1 - 2*y + y**2 + 8*x**2)**S.Half: 1,
S.Half + S.Half*y - S.Half*(1 - 2*y + y**2 + 8*x**2)**S.Half: 1,
}
assert roots(a*b*c*x**3 + 2*x**2 + 4*x + 8, x, cubics=False) == {}
assert roots(a*b*c*x**3 + 2*x**2 + 4*x + 8, x, cubics=True) != {}
assert roots(x**4-1, x, domain='Z') == {S.One: 1, -S.One: 1}
assert roots(x**4-1, x, domain='I') == {I: 1, -I: 1}
assert roots((x-1)*(x+1), x) == {S.One: 1, -S.One: 1}
assert roots((x-1)*(x+1), x, predicate=lambda r: r.is_positive) == {S.One: 1}
assert roots(x**4-1, x, domain='Z', multiple=True) == [S.One, -S.One]
assert roots(x**4-1, x, domain='I', multiple=True) in ([I, -I], [-I, I])
assert roots(x**3, x, multiple=True) == [S.Zero, S.Zero, S.Zero]
assert roots(1234, x, multiple=True) == []